Ring Structures of Metal Atom Doped C9, C11, and C13 Clusters from Ab Initio Calculations—The Finding of a Gd@C13 Magnetic Superatom Ring

Pure C9 clusters have a linear chain structure. However, here, we report using ab initio calculations the transformation of a chain into a cyclic ring structure with the capping of Ca, Sr, and Ba atoms. Further calculations on neutral and charged clusters doped with Sc, Y, and La atoms show stabilization of a cation C9 isoelectronic cyclic ring capped with the metal (M) atom, but anion clusters doped with these trivalent atoms form a C10 like MC9 ring, which deforms to a necklace structure. Both the ring structures correspond to electronic shell closing with 20 delocalized valence electrons in a disk jellium model and have a large highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) gap. Calculations of IR and Raman spectra show no imaginary frequency, suggesting that the structures are stable. Addition of two C atoms to the ring structure of LaC9 leads to a capped ring structure of the cation LaC11 and an open ring structure of the LaC11 anion. Further addition of two C atoms leads to a La@C13 cation as well as Ca@C13 and Sr@C13 neutral wheel-shaped rings with endohedral doping of the M atom. These novel ring structures have a large HOMO–LUMO gap of more than 4 eV and are magic with electronic shell closing corresponding to 28 delocalized valence electrons in a disk jellium model. There is π aromaticity in this ring satisfying 4n+2 (n = 3) Hückel’s rule with 14 valence electrons. Interestingly, when the dopant is a Gd atom, there is a formation of a magnetic superatom ring Gd@C13+ with 7 μB magnetic moments due to seven 4f up-spin states of Gd being fully occupied. Bonding in these novel ring structures is discussed.


INTRODUCTION
Clusters and nanostructures of carbon have attracted great interest in the past few decades with path breaking discoveries of fullerenes, 1 nanotubes, 2 and graphene. 3Doping of metal (M) atoms in these nanostructures has also attracted much interest in manipulating their properties for different applications including catalysis, sensors, and functionalized materials.This has led to the discovery of endohedral fullerenes 4 with an M atom or a group of atoms inside a fullerene cage.In these doped species, generally, no significant change occurs in the atomic structure of the host cluster or the nanostructure due to doping.However, in other related systems such as clusters of silicon and Ge, the doping of an M atom led to the finding of novel structures 5−8 with very different atomic arrangements compared with those of the bulk or undoped clusters.Even the formation of nanotubes 9−11 of silicon has been shown with the doping of M atoms.An interesting situation arises when the number of carbon atoms in a fullerene reduces to less than 30.With the shrinking of the cage, the bonding develops significant sp 3 character due to the reduction in the number of hexagons, and in the smallest fullerene C 20 with all pentagonal faces, the bonding becomes close to sp 3 type.This leads to lower stability of the cage due to the dangling bonds.It has been found 7,12−14 that the stability of such small carbon fullerenes with 20−28 atoms can be enhanced with the doping of an M atom as in the case of endohedrally doped silicon fullerenes and other polyhedral structures. 15In these cases, the M atom interacts strongly with the silicon and carbon fullerene cages.We ask the following question: what will happen to the structures if the number of carbon atoms is reduced further to less than 20 and if their structures can be manipulated with the doping of an M atom?
Carbon clusters with less than 20 atoms have a sudden change in the structure with rings becoming more favorable. 16,17urrently, these systems are attracting much interest with reports of the formation of C 10 , C 14 , C 16 , and C 18 rings 18−20 on a substrate.Clusters with less than 10 carbon atoms and particularly those with an odd number of atoms such as C 7 and C 9 have a chain structure. 21Doping of an atom in some small clusters of carbon such as CuC 10 , SiC 9 , and so on has been studied, 22,23 but the structures of the doped clusters remain similar to those of the pristine ones.In a study 24 on La doped carbon cation clusters, mobility experiments suggested the formation of M atom doped ring structures with varying number of carbon atoms, although no theoretical understanding or confirmation was achieved.It is of interest to explore the doping of M atoms in small carbon clusters − an area that is still not well studied − and to search if there could be new forms of carbon.Very recently, doping of an M atom (M = Be, Mg, and Al) in the C 9 cluster has been shown 25 to transform a chain structure of C 9 to a ring structure.Here, we study doping of a C 9 cluster with Ca, Sr, and Ba atoms and show that these doped clusters favor a ring structure of carbon with the M atom capping the ring in contrast to M = Be and Al for which the M atom is part of the ring.The choice of a divalent atom emerged from the fact that the LUMO and LUMO+1 states of a C 9 ring have large energy gap.The LUMO can be occupied if a divalent electropositive atom is added, leading to the possibility of a highly stable doped C 9 ring.Furthermore, we considered doping of Sc, Y, and La atoms as well as their anion and cation species.Our results show that the cations of Sc, Y, and La doped clusters are isoelectronic to those doped with a divalent M atom and have a cyclic ring structure with the M atom capping the ring, but the anion of all the three trivalent M doped clusters stabilizes in another ring structure in which the M atom is incorporated in a ring similar to a highly stable C 10 ring.In fact, the anions of these doped clusters become electronically equivalent to C 10 with the same number of valence electrons, but the optimized ring relaxes to a necklace shape.By further adding two C atoms, we formed a MC 11 (M is a trivalent atom) ring, which is favored by the cation, but for the La doped anion, the ring becomes open.Therefore, we added two more C atoms and found a novel La@C 13 cation cyclic ring with the La atom endohedrally doped inside the C 13 ring.Similar results have been obtained for the isoelectronic Ca and Sr doped C 13 neutral rings, while Ba sits slightly above the ring.There is a large HOMO−LUMO gap in these cases suggesting the chemical stability of these rings with a closed electronic shell.As rare earth atoms also behave as trivalent, we further considered M as a Gd atom.Interestingly, it forms a Gd@C 13 + magnetic superatom ring with half-filled 4f levels and 7 μ B magnetic moments.

RESULTS AND DISCUSSION
The optimized chain and ring structures of the pure C 9 cluster are shown in Figure 1.The chain structure lies 25 0.435 eV (0.250 eV) lower in energy compared with a ring structure using PBE (PBE0).There is a small variation in the C−C bond lengths (1.296 Å at the ends, 1.295, 1.276, and 1.281 Å in the middle of the chain using PBE as shown in Figure 1), and the magnetic moment is zero.The bond lengths become slightly contracted (1.283 Å at the ends, 1.286, 1.267, and 1.273 Å in the middle) when PBE0 is used.The HOMO−LUMO gap is 1.715 eV (2.978 eV) using PBE (PBE0).The ring structure is slightly buckled and there is a small variation in the nearest neighbor C− C bond lengths, which lie in the range of 1.301−1.329Å (1.286−1.322Å) using PBE (PBE0).These results show a slight increase in the bond lengths in the ring isomer compared with the chain.The ring isomer also has 0 μ B magnetic moment, and the HOMO−LUMO gap is smaller [0.648 eV (2.378 eV) using PBE (PBE0)] compared with that of the chain.It is also much smaller than 3.750 eV (5.70 eV) obtained for a C 10 5-fold symmetric ring within the PBE (PBE0).The large value of the HOMO−LUMO gap for C 10 makes it a particularly stable ring.All of the nearest neighbor bond lengths for C 10 are equal to 1.295 (1.288 Å) using PBE (PBE0) and are slightly smaller than the values for C 9 .We also calculated the C 9 anion using the Gaussian code, and the chain structure is found to be 1.016 eV lower in energy than a ring isomer.This result agrees with experiments 16 where a chain of C 9 anion has been reported.For C 11 and C 13 , the converged chain and ring structures are also shown in Figure 1 and the ring structures are 0.935 and 1.124 eV lower in energy, respectively, than the chain structure using PBE.The C 13 ring has 2 μ B magnetic moments while the C 11 ring is nonmagnetic.The bond lengths in C 11 ring lie in the range of 1.255−1.352Å using PBE (see Figure 1) while the HOMO− LUMO gap is 0.57 eV.However, using PBE0, the HOMO− LUMO gap is 2.114 eV.The atomic structure of the C 11 ring can be derived from a C 12 hexagon by removing an atom.For C 13 , the ring is symmetric.All of the bond lengths are equal (1.291 Å using PBE) and are slightly shorter than those of C 10 .The HOMO−LUMO gap is 0.246 eV within PBE, but it is 2.055 eV using PBE0.The C 13 chain isomer, however, shows variation in the bond lengths (Figure 1).We find an oscillatory behavior of the bond lengths in chain isomers.We expect only a small difference in the bond lengths calculated by using PBE and PBE0, but the HOMO−LUMO gap is significantly underestimated in PBE.
We further added an M atom at one of the ends of the C 9 chain, capped the C 9 ring, and inserted it in the ring to make a MC 9 ring.The optimized structures for the M = Ca, Sr, and Ba atom doped systems are shown in Figure 2. It is found that for these M atoms, the capped ring (Figure 2a−c) structure has the lowest energy.This is different from the case 25 of M = Be and AlC 9 cation cluster.There is a small variation in C−C bond lengths in the C 9 ring, which vary in the range of 1.27−1.34Å for For C 9 , the chain isomer lies 0.435 eV lower in energy than the ring isomer, while for C 11 and C 13 , the ring isomer lies 0.935 and 1.124 eV lower in energy, respectively, than the chain isomer using PBE.For the ring isomers, we have also shown the side view.It is seen that the ring isomer of C 9 is buckled, but for C 11 and C 13 , all atoms lie in a plane.The C 13 ring has 2 μ B magnetic moments and all the bond lengths are equal (1.291 Å), while in other cases, there are small variations and the magnetic moment is zero.The bond lengths are given in Å. M = Ca, while the C−Ca bond lengths vary in the range of 2.41− 2.48 Å using PBE.There is a kind of pentagonal distortion in the ring with a mirror symmetry in a plane perpendicular to the ring.As there is a large gap between LUMO and LUMO+1 for the ring isomer of C 9 , the addition of a divalent electropositive M atom fills the LUMO due to the charge transfer (see below) and leads to a large HOMO−LUMO gap of 4.046, 3.821, and 3.908 eV for M = Ca, Sr, and Ba, respectively, in the capped ring isomer using PBE0 in the Gaussian code.Very similar values (4.153, 3.846, and 3.902 eV, respectively) of the HOMO−LUMO gap have been obtained using VASP with PBE0.This reflects the good accuracy of our calculations.There are large dipole moments on these capped rings, namely, 6.76, 8.03, and 9.08 D, respectively, for M = Ca, Sr, and Ba due to the charge transfer from the M atom to the C 9 ring.These doped clusters have 38 valence electrons.Among these, two valence electrons in each carbon atom are covalently bonded with neighboring carbon atoms (with some deviation from a chain).This takes away 18 valence electrons from the C atoms.The remaining 20 delocalized valence electrons participate in bonding between the carbons and the M atom.As this delocalized charge lies mostly in the ring and between the ring and the M atom, we can consider a jellium model for a circular disk 26 according to which clusters with 20 valence electrons correspond to an electronic shell closing and are magic.Therefore, the large HOMO− LUMO gap in these capped clusters is due to an electronic shell closing leading to the high stability of these species as it was also found 25 in the case of a Be doped C 9 ring cluster.Furthermore, it was shown that this ring cluster has π aromaticity with 10 valence electrons corresponding to the 4n+2 (n = 2) Huckel rule.We analyzed the energy levels and the molecular orbitals (MOs) in the case of CaC 9 , and its energy spectrum with MOs is shown in Figure S1.It is found that the spectrum has similarity with that of the Be@C 9 case, 25 and it also satisfies Huckel's π aromaticity rule of 4n+2 (n = 2) corresponding to 10 valence electrons.
On the other hand, substitution of a divalent atom in place of a C atom in a C 10 ring leads to a necklace shaped structure (Figure 2d−f) with 2 μ B magnetic moments, which lie 1.199 (0.743), 0.907 (0.020), and 0.843 (0.173) eV higher in energy, respectively, for Ca, Sr, and Ba than the capped ring isomer using PBE (PBE0) in VASP.All the atoms in the ring lie nearly in a plane with small deviations.The chain isomer lies 2.072 (Ca), 1.820 (Sr), and 1.636 eV (Ba) higher in energy using PBE than the capped ring isomer.In the C 10 ring isomer, 20 valence electrons are used in σ bonding between the neighboring C atoms while the remaining 20 delocalized valence electrons again form an electronic closed shell and make it magic.Accordingly, MC 9 with M as a divalent atom has 38 valence electrons, which are two electrons short of electronic shell closing leading to 2 μ B magnetic moments.Therefore, clusters with 38 valence electrons may favor a capped ring structure, while those with 40 valence electrons may favor a ring structure with M incorporation.Deviation from this number of valence electrons could give rise to magnetic moments on these clusters.We also tried capping an Mn atom on a C 9 ring, and the optimized structure shows it to have 5 μ B magnetic moments due to the five 3d unpaired electrons while the remaining two valence electrons occupy the LUMO of the C 9 ring.The optimized structure is similar to that shown in Figure 2a.However, the other ring isomer with the Mn atom in the ring lies 1.015 eV lower in energy, and it has 3 μ B magnetic moments as four valence electrons on the Mn atom are used in bonding in the ring.The atomic structure is like that shown in Figure 2d.The magnetic moments are mostly localized on the Mn site, which has 4.27 μ B , while four C atoms have about 0.26 μ B magnetic moment each and five C atoms have −0.42 to −0.53 μ B magnetic moment using PBE.There is a reflection symmetry in a plane bisecting the necklace in two halves perpendicularly and passing through the Mn atom.The Mn−C bond length is 1.914 Å, while the C−C bond lengths are 1.273, 1.323, 1.277, and 1.304 Å.Within the ring, there is antiferromagnetic coupling with −0.43, 0.26, −0.46, 0.25, −0.54, 0.26, −0.46, 0.26, and −0.43 μ B magnetic moments cyclically on the carbon atoms.The HOMO−LUMO gap is 1.0 eV (2.732 eV) by using PBE (PBE0).
Further calculations on cations of rings doped with Sc, Y, and La atoms show that these species being isoelectronic to Ca, Sr, and Ba doped C 9 , respectively, favor capping of the C 9 ring by the M atom as shown in Figure 3.In the case of ScC 9 cation, the ring becomes very nearly circularly symmetric (Figure 3a) with the mean C−C and C−Sc bond lengths of 1.307 and 2.227 Å, respectively, while for the YC 9 (LaC 9 ) cation, the nearest neighbor C−C and C−M bond lengths lie in the ranges of 1.287−1.325Å (1.274−1.338Å) and 2.377−2.408Å (2.546− 2.598 Å), respectively, using PBE0 in Gaussian program.The atomic structures remain similar when optimized using PBE and PBE0 except for a small contraction in the bond lengths when PBE0 is used.There are large HOMO−LUMO gaps of 4.601, 4.607, and 4.547 eV for ScC 9 + , YC 9 + , and LaC 9 + , respectively, using PBE0.The cation of Sc, Y, and La doped C 10 like ring lies 2.738, 2.021, and 1.694 eV higher in energy than the capped ring isomer using PBE0 in the Gaussian program.The C 10 like ring relaxes to a necklace structure and has 2 μ B magnetic moments.Note that the MC 9 + (M is a trivalent atom) ring has two valence electrons less than those in C 10 , which gives rise to 2 μ B magnetic moments and a smaller HOMO−LUMO gap.On the other hand, anions of Sc, Y, and La doped species interestingly favor the C 10 like ring (see Figure 4) and they lie 0.185, 0.716, and 0.885 eV lower in energy than the capped C 9 ring isomers, respectively.These results show that the stability of the necklace isomer increases as one goes down in the group IIIA column in the periodic table.In both C 10 neutral and MC 9 (M = trivalent atom) anion cases, the number of valence electrons is the same.The M doped carbon ring relaxes to a necklace shape (Figure 4a) or a butterfly shape (Figure 4b,c 4. All the lowest energy isomers have no imaginary frequency and are dynamically stable.This has been further checked for the CaC 9 case using ab initio molecular dynamics calculations by heating to 400 K, and the capped ring structure does not change much except for some displacement of atoms.It is to be further noted that the necklace shaped ring isomer has two structures: 1) with the M atom lying in the ring as shown in Figure 3f for LaC 9 + where the M atom interacts with two C atoms and 2) where the M atom drifts inside the ring and interacts with more than two C atoms as shown in Figure 3d,e for M = Sc and Y, respectively, as well as in Figure 4a−c.
As the size of the M atom is large, the ring of carbon atoms is partial in some cases, i.e., open, and the M atom interacts with many C atoms.These results suggest that we can add C atoms to complete the carbon ring.We added two C atoms in the case of LaC 9 to form LaC 11 , and the optimization of the cation using PBE0 in Gaussian code shows (Figure 4g) that a structure with La atom in the ring has the lowest energy while a capped ring (Figure 4h) lies slightly lower in energy using PBE in VASP.Both have 2 μ B magnetic moments.In the case of the LaC 11 anion, the carbon ring is open and distorted as shown in Figure 4i.Therefore, La is still too large for the C 11 ring to be accommodated inside it.Subsequently, we added two more C atoms to form an M@C 13 ring in which the La atom is well accommodated endohedrally as shown in Figure 5a.This beautiful wheel-shaped cation cluster is a singlet as the two valence electrons from La + occupy the two down spin states of C 13 giving rise to a large HOMO−LUMO gap.The C−C bond lengths are nearly the same as in the undoped C 13 ring, and therefore, there is little distortion from the doping of the M atom.This M doped ring has effectively 28 delocalized valence electrons while 26 valence electrons on C atoms are involved in σ bonding between the neighboring C atoms.Clusters with 28 delocalized electrons are magic in a disc jellium model 26 and are more stable than the neighboring sizes.Accordingly, this ring should exist in a high abundance.We find that there is a large   4g) with the La atom in the ring has the lowest energy with PBE0 in Gaussian, but using PBE in VASP, the capped ring (h) lies slightly lower in energy.In Gaussian, the capped ring structure becomes distorted and lies slightly higher in energy.Both the top and side views are shown.One can see from the side view that the ring becomes slightly distorted in the capped ring isomer for the MC 9 anion, while for the LaC 11 anion (i), the ring is open.
HOMO−LUMO gap of 2.770 eV (2.908 eV) within PBE0 using VASP (Gaussian), which suggests this ring to be good for visible light emission.The C−C bond lengths are all equal to 1.289 Å (1.291 Å), while C−La bond lengths have the value 2.694 Å (2.699 Å) in VASP (Gaussian).All of the vibrational frequencies are real, and this suggests that the ring structure is dynamically stable.Our results are in general agreement with the finding 24 of ring isomers of La doped cation clusters.We further doped Ca, Sr, and Ba atoms to form neutral Ca@C 13 , Sr@C 13 , and Ba@C 13 rings.The converged ring structures are also circularly symmetric (Figure 5b), but Ba lies above the ring due to its large atomic size.There is a large HOMO−LUMO gap of 4.448, 4.410, and 4.317 eV, respectively, suggesting their chemical stability.
In Figure 6, we have shown the electronic energy levels and the MOs of the Ca@C 13 ring.Considering the z axis to be normal to the ring, the ordering of the occupied MOs is 1S, 1P .Accordingly, there are seven π-bonded MOs with 14 valence electrons.Therefore, this ring satisfies the 4n+2 (n = 3) Huckel's rule of aromaticity.We have also shown the angular momentum and site projected density of states obtained from the VASP calculations using PBE0 for this ring in Figure S2.One can see that the low-lying states arise from the 2s atomic orbitals of carbon atoms while the higher lying states arise from the 2p atomic orbitals of carbon atoms.There is no significant contribution from the Ca atom in the occupied region, suggesting nearly complete charge transfer to the carbon ring.
We further doped the C 13 ring with a Gd atom and the Gd@ C 13 cation ring remains stable as shown in Figure 5c although the Gd ion gets very slightly displaced from the center of the ring using PBE.The displacement from the center is more significant when PBE0 is used (Figure 5d).This also creates some variation in the C−C bond lengths, which lie in the range of 1.282−1.294Å.The C−Gd bond lengths are in the range 2.454−3.030Å.There is a large magnetic moment of 7 μ B , which is mostly localized on the Gd ion (see Figure 7i) as the seven up spin 4f electrons on Gd remain unpaired except for some hybridization effects with the carbon ring.The up-spin and down-spin densities of states of this ring are also shown in Figure S2.
Comparing this with the density of states for Ca@C 13 (Figure S2), one finds that the spectrum of Gd@C 13 is similar to that of Ca@C 13 except for the seven 4f up-spin states of Gd while the down-spin 4f states lie above the HOMO.The doubly degenerate spin states at −4.14 eV have 2D character, and there is hybridization of the 5d atomic orbitals of Gd with the 2D MOs of the ring resulting in a shift to higher binding energies compared to the case of Ca@C 13 but the hybridization with other occupied states is quite small.There is a large HOMO− LUMO gap of 3.353 eV within PBE0, and therefore, we call it a magnetic superatom ring.There is about 1.2 e of charge transfer from Gd to the carbon ring.Such magnetic superatoms have been predicted 27,28 earlier for Mn@Sn 12 , Gd@Au 15 , and some other cases. 7They should also exist with different spins if other rare earth atoms are doped similar to the case of the doped silicon clusters. 29n Figure 7, we have shown the charge density isosurface for a few representative cases such as the CaC 9 capped ring (Figure 7a), YC 9 + capped ring (b), YC 9 − C 10 -type ring (d), LaC 11 + capped ring (f), and La@C 13 + endohedral ring (g) using VASP  with PBE.Also, in Figure 7c,e,g, we have shown the charge density isosurface for the YC 9 + capped ring, YC 9 − open ring, and LaC 13 cation with a higher value (0.1, 0.05, and 0.05 e/Å 3 , respectively) of the isosurface.It is seen that there is a low charge density between the ring and the M atom.For the CaC 9 case, there is little valence charge on the Ca ion due to the charge transfer to the C 9 ring while high density of charge is there between the C ions.Note that we have considered 3p core electrons of Ca as valence electrons, and therefore, some charge is seen around the Ca ion.Similar results have been obtained for Sr and Ba capped rings as well as for the YC 9 cation (Figure 7b,c).For the YC 9 anion ring isomer also, there is covalent bonding between the neighboring C atoms and low charge density between the C atoms and M atom (Figure 7e).The remaining valence electrons are delocalized on the necklace and between the carbon necklace and the Y ion.Furthermore, in the case of La@C 13 + , the ring is nearly circular, and as shown in Figure 7g,h, there is covalent bonding between the nearest neighbor C atoms with high charge density so that one can consider 26 valence electrons to take part in it.The remaining 28 valence electrons are delocalized on the ring and between the ring and the M atom.In the case of Gd@C 13 + , the bonding character is similar to that in the case of La or Ca doped ring, but there are seven unpaired 4f electrons that give rise to spin polarization (Figure 7i), which can be seen to be mostly localized on the Gd ion.We further calculated the interaction of a ring with an M atom and the energy gained by the doping of an M atom, E DE , has been calculated from where E C and E M are the energies of the carbon ring/chain and the M atom, respectively, while E S is the energy the M atom doped system.We find that there is a gain of 5.009, 4.682, 5.193, and 3.562 eV for the doping of Ca, Sr, Ba, and Mn atoms compared with the ground state of C 9 , the chain isomer.This energy gain further to 6.296, 6.292, and 7.206 eV for the doping of Sc, Y, and La atoms, respectively, within PBE.For the C 13 ring, the doping energy becomes 5.991, 5.953, 6.348, 6.826, 7.176, and 8.083 eV for the doping of Ca, Sr, Ba, Sc, Y, and La atoms, respectively.The large gain in energy is responsible for the transformation of a C 9 chain to a ring as well as further stabilization of the C 13 ring.The adiabatic ionization potentials of M@C 13 (M = Sc, Y, and La) are 6.474, 6.109, and 5.444 eV while the adiabatic electron affinities are 1.722, 1.631, and 1.873 eV, respectively.Large ionization potentials and low electron affinities are characteristic of magic clusters.Bader charge analysis of the clusters was performed to further understand the bonding character in these doped rings.In the case of Ca, Sr, and Ba capped C 9 rings, there is ∼1.5 e charge transfer from the M atom to the carbon ring.The charge is not equally distributed on all of the C atoms in the ring.Three C atoms have about 0.66 e excess charge, while one C atom has 0.34 e excess charge.The remaining five C atoms give about 0.09 to 0.32 e to neighboring C atoms.Therefore, there is a weak ionic bonding character within the ring and strong ionic bonding between the M atom and the ring.A similar behavior is seen for the cations of the Sc and Y capped rings.For ScC 9 + , there is 1.82 e charge deficiency on Sc while seven C atoms have small excess of charge (0.04 to 0.21 e), and two C atoms have a slight (0.02 and 0.1 e) deficiency of charge.Accordingly, the charge is dominantly removed from Sc for the cation, and the charge transfer to the ring is less compared with the doping of divalent atoms.Similar results are obtained for YC 9 + .In this case, there is a deficiency of 1.95 e on Y while a small charge (0.04−0.24 e) is transferred to seven C atoms and one C atom has a very slight depletion of charge (0.01 e).In these cases, about one valence electron remained on the metal ion.These effectively 20 delocalized valence electron species are also magic clusters.
In Figure 8, we have shown the Gaussian broadened IR and Raman spectra of CaC 9 as a representative of the capped ring structure.The IR spectrum has two intense peaks at 1807.78 and  1866.08 cm −1 , which correspond to the stretching of the C−C bonds.There are two weaker peaks at 1096.62 and 1101.12cm −1 , which correspond to bond stretching and the swing of the ring.The small peak at 297.99 cm −1 corresponds to breathing of the ring and up−down motion of the Ca atom.The Raman activity shows a peak at 225.81 cm −1 , which is a bending mode, while a peak at 872.39 cm −1 corresponds to the breathing mode of the ring.The strongest peak at 1478.62 cm −1 arises from the C−C bond stretching.Another peak at 1866.08 cm −1 also corresponds to the C−C bond stretching.Similar results were obtained for SrC 9 and BaC 9 .The IR spectrum for SrC 9 has strong peaks at 1816.74 and 1839.65 cm −1 and two almost degenerate peaks at 1093.10 and 1093.43 cm −1 .As these modes arise from the motion of carbon ions, the frequencies are similar in the case of CaC 9 .While the Raman activity shows a strong peak at 1507.88 cm −1 corresponding to C−C bond stretching.Another strong peak at 856.05 cm −1 corresponds to the breathing mode of the ring.The smaller peak at 229.76 cm −1 is the bending mode of the ring.For BaC 9 , the corresponding IR peaks are at 1820.46, 1852.73,1092.57,1093.38, and 222.13 cm −1 while the Raman activity peaks are at 1502.08, 854.23, and 224.64 cm −1 .In the case of the YC 9 cation, there are two prominent IR modes: 1) at 304.11 cm −1 and 2) a doubly degenerate mode at 1101.75 cm −1 , which corresponds to C−C bond stretching.The Raman activity has a strong peak at 875.98 cm −1 corresponding to the breathing mode and two weaker peaks at 1522.69 and 1525.92cm −1 corresponding to C−C bond stretching.For the LaC 11 cation ring isomer (Figure 4g), the IR spectrum has strong peaks at 1938.78 and 1995 cm −1 both corresponding to C−C bond stretching and weaker peaks at 1666.88 and 1132.91 cm −1 corresponding to C−C bond stretching, and a peak at 321.89 cm −1 corresponding to the swinging motion of the ring.There is one very strong Raman active mode at 1938.78 cm −1 corresponding to C−C bond stretching.Other modes are much weaker.These results will be helpful in identifying the correct isomer in experiments.
The IR spectrum and Raman activity of the Ca@C 13 ring are shown in Figure 9 as a representative of this ring structure.The high symmetry of the ring leads to only a few intense lines in the spectra.The IR spectrum has a degenerate strong mode at 860.25 cm −1 , which is the stretching and swinging mode of the ring and does not involve motion of the M atom.There is a mode at 99.18 cm −1 , which is the breathing mode of the M doped ring, while a doubly degenerate mode at 78 cm −1 is the swinging mode of the M atom and the ring.The highest Raman intensity at 647.24 cm −1 corresponds to the breathing mode of the ring.The doubly degenerate mode at 118.74 cm −1 is the bending mode of the ring while the doubly degenerate mode at 1278.23 cm −1 is the stretching mode of the ring.In all these modes, the M atom remains at its position.

SUMMARY
In summary, we have reported the stabilization of ring structures of C 9 with the doping of a metal atom.Divalent atoms such as Ca, Sr, and Ba as well as isoelectronic cations of Sc, Y, and La favor a capped C 9 ring structure, while the anions of these trivalent atom doped clusters stabilize in a ring structure of MC 9 − where the metal atom is part of the ring.These species correspond to an electronic shell closing with 20 delocalized valence electrons in a disk jellium model and have large HOMO−LUMO gaps.Also, following the Huckel 4n+2 aromaticity rule, there is π aromaticity with 10 valence electrons (n = 2).Further studies of the doping of a La atom have led to the findings of a necklace shaped ring of the LaC 11 cation as well as a La@C 13 ring cation with endohedral doping of a La atom in a wheel shaped C 13 ring.The La@C 13 cation ring has 28 delocalized valence electrons, which also correspond to an electronic shell closing in a disc jellium model.This ring also has π aromaticity with n = 3.Our results agree with the experimental report of La doped ring structures of carbon cation clusters in this size range and show for the first time their most favorable structures.Similar results have been obtained for Sc and Y doped cation clusters as well as for the divalent atom (Ca, Sr, and Ba) doped neutral clusters.We hope that these novel structures will stimulate further research on metal atom doped carbon clusters and other nanostructures.These species would be interesting for the study of single atom catalysis − a subject that is drawing great interest in recent years − as well as for understanding extended systems such as interaction of metal atoms with a defective graphene.Also, we found a magnetic superatom Gd@C 13 + ring with 7 μ B large magnetic moments (all up spin 4f states of Gd occupied) and a large HOMO−LUMO gap in which the Gd atom is endohedrally doped in a C 13 ring.This could give a new direction to make the smallest magnetic species.Earlier magnetic slaved atoms have been shown in hydrogenated silicon cages 30 and rare earth atom doped silicon cages with different magnetic moments. 29As the size of rare earth atoms are similar, we expect magnetic ring structures of carbon to have different magnetic moments with the doping of other rare earth atoms, and that our results will encourage researchers to find such species in experiments and further development of carbonbased smallest magnetic nanostructures.

COMPUTATIONAL METHOD
The calculations have been performed using projector augmented wave pseudopotentials 31 in Vienna Ab initio Simulation Package 32 (VASP) and spin-polarized generalized gradient approximation 33 of Perdew, Burke, and Ernzerhof

Figure 1 .
Figure1.Optimized atomic structures of chain and ring isomers of C 9 , C 11 , and C 13 clusters.For C 9 , the chain isomer lies 0.435 eV lower in energy than the ring isomer, while for C 11 and C 13 , the ring isomer lies 0.935 and 1.124 eV lower in energy, respectively, than the chain isomer using PBE.For the ring isomers, we have also shown the side view.It is seen that the ring isomer of C 9 is buckled, but for C 11 and C 13 , all atoms lie in a plane.The C 13 ring has 2 μ B magnetic moments and all the bond lengths are equal (1.291 Å), while in other cases, there are small variations and the magnetic moment is zero.The bond lengths are given in Å.
Similar results have been obtained for Sr (Ba) with the C−C bond lengths lying in the range of 1.27−1.35Å (1.28−1.35Å) and Sr−C (Ba−C) bond lengths lying in the range of 2.57−2.65 Å (2.74−2.83Å).It is noted that the change in C−C bond lengths is quite small, but M−C bond lengths increase significantly in going from M = Ca to Sr and then to Ba due to the increasing size of the M atom.

Figure 2 .
Figure 2. Capped ring and ring (necklace) structures of (a,d) CaC 9 , (b,e) SrC 9 , and (c,f) BaC 9 , respectively.The capped ring structures have the lowest energy.Both the top and side views are shown.For Mn doping, the structures are similar to those in Figure 2a,d with the ring isomer lying lower in energy than the capped ring.The large (small) balls show M (C) atoms in each case.
), and the M atom drifts inside the ring (Figure 4a−c).There are large HOMO−LUMO gaps of 2.957, 3.114, and 2.726 eV, respectively, for M = Sc, Y, and La doped anions using PBE0, and in all cases, the magnetic moment is zero.The HOMO−LUMO gap is smaller than the values for the capped ring cations.The substitutional doping of an M atom in C 10 reduces its symmetry as well as the HOMO− LUMO gap (5.70 eV) and brings the latter to the visible range.Accordingly, these M doped species become optically interesting.The C−C (C−M) bond lengths vary in the range of 1.271−1.356(2.206−2.328Å), 1.258−1.381(2.404−2.579Å), and 1.259−1.376Å (2.556−2.736Å) for MC 9 anions, where M = Sc, Y, and La, respectively.The carbon ring in the capped ring isomer is slightly distorted, as shown in Figure

Figure 3 .
Figure 3. Capped ring and ring structures of cations of (a,d) ScC 9 , (b,e) YC 9 , and (c,f) LaC 9 , respectively.The capped ring structure has the lowest energy.Both the top and side views are shown.The large atom is the M atom.

Figure 4 .
Figure 4. Ring and capped ring structures of anions of (a,d) ScC 9 , (b,e) YC 9 , (c,f) LaC 9 , (g,h) LaC 11 cation, and (i) LaC 11 anion.The ring structures have the lowest energy for the MC 9 anions.For the LaC 11 cation, the ring structure (Figure4g) with the La atom in the ring has the lowest energy with PBE0 in Gaussian, but using PBE in VASP, the capped ring (h) lies slightly lower in energy.In Gaussian, the capped ring structure becomes distorted and lies slightly higher in energy.Both the top and side views are shown.One can see from the side view that the ring becomes slightly distorted in the capped ring isomer for the MC 9 anion, while for the LaC 11 anion (i), the ring is open.

Figure 5 .
Figure 5. Optimized structure of (a) the La@C 13 cation using PBE0 in VASP.It remains very similar when PBE is used.The side view shows that La lies in the plane of the ring.Similar results were obtained by using Gaussian and PBE0.(b) The atomic structure for neutral Ca@C 13 and Sr@C 13 as well as for Sc and Y doped cations in Gaussian with PBE0.The bonds between the M and C atoms have not been connected.(c) The cation of Gd@C 13 with Gd at the center of the ring and in the plane of the ring.This is obtained when PBE is used in the VASP.(d) Optimized structure when PBE0 is used in VASP.The Gd ion is slightly displaced from the center though it lies in the plane of the ring.

Figure 8 .
Figure 8. IR and Raman activity spectra for the Ca capped C 9 ring cluster CaC 9 with a Gaussian broadening of half width at a halfmaximum of 4 cm −1 .

Figure 9 .
Figure 9. IR and Raman activity spectra for the Ca doped C 13 ring cluster Ca@C 13 with a Gaussian broadening of half width at a halfmaximum of 4 cm −1 .